How do you prove that every galilean transformation of the space $\mathbb R \times \mathbb R^3$ can be written in a unique way as the composition of a rotation, a translation and uniform motion? Thanks!
As mentioned, this problem appears in V.I Arnold's "Mathematical Methods of Classical Mechanics".
A simple proof can be found on page 7 of this nice lecture series on the topic. In essence, you use the affine property of the transformation to show that the time cannot be dilated, the space component must be multiplied by a orthogonal matrix plus a constant vector. The uniqueness follows by considering to different parameterizations and proving equality.